# Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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### Is the set of irrational numbers smaller than the set of real numbers? [duplicate]

I am not a mathematician, but I study it for fun. I know that Cantor showed that the infinite set of the real numbers is larger than the rationals since the real numbers are uncountable and the ...
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### a name for a class of cardinals

I'm not a native English speaker; what S in SCar below might stand for ?
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### Proving that $\left|A \times A\right|$ is equal to $\left|A\right|$ for every infinite set

How do you prove that $\left|A \times A\right|$ is equal to $\left|A\right|$ for every infinite set? I'm trying to prove this basic fact of cardinal arithmetic, but I'm getting stuck on the ...
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### Is there an intuitive way of justifying why the square of an infinite cardinal is itself?

By no means I am an expert in this subject, but I do have some knowledge of ZFC. While there are many proofs which are difficult to recollect, I feel like I have enough knowledge that if I am given a ...
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### How to prove $a^x\times a^y=a^{x+y}$ for cardinals? [duplicate]

How can I prove this: $$a^x\times a^y=a^{x+y}$$ when $card(A)=a$ , $card(X)=x$ and $card(Y)=y$.
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### Proof with transfinite induction

I'm trying to prove the following statement: Suppose that for every $r\in\mathbb{R}$ we are given a finite set $A_r\subseteq\mathbb{R}$ and that for any finite set $D\subseteq\mathbb{R}$, there ...
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### What is aleph-one to the power of aleph-null

What is $\aleph_1^{\aleph_0}$? Can anyone shed some light. I'm not sure if this is provable or even has a value. Thanks
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### The set of polynomials with $n$ variables and coefficients in a countable set is countable

I'd like to know whether the set of polynomials $S[X_1, \dotsc, X_n]$ with coefficients in a countable set $S$ is countable. I think it must be, but I don't know how to prove it. Thanks.
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### Let $G$ be the group of all the maps from closed interval $[0,1]$ to $\mathbb{Z}$.

Let $G$ be the group of all the maps from closed interval $[0,1]$ to $\mathbb{Z}$. The subgroup $H= \left \{ f \in G :f(0)=0 \right \}$ Then $1)$ $H$ is countable $2)$ $H$ is uncountable $3)$ $H$ ...
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### Explanation of Cardinal Arithmetic Used in Proving that bases of Vector spaces have the same cardinality.

Let $V$ be a vector space with bases $B_1$, $B_2$. For all $b\in B_1$ there exists $U_b\subset B_2$ such that $U_b$ is finite and $b\in span(U_b)$. Hence, $V=span(B_1)=span(\cup_{b\in B_1}U_b)$. Since ...
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### $(\lambda,D)$-model homogenity

Here on the page $41$ in the definition $6.1(3)$ I do not follow where $D$ from $6.1(3)$ appears in the definition of $(\lambda,D)$-model homogenity in $6.1(2)$. It appears in the first $2$ paragraphs ...
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### Unknown index in a model theoretic considerations

What is $k$ in $${}^kM$$ here on the page $11$ in the definition $2.10$?
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### Uncoutable set minus a countable subset is uncountable without axiom of choice or continuum hypothesis [duplicate]

Is there a way of proving that if $|X|=2^{\aleph_0}$ and $Y\subseteq X$ such that $|Y|=\aleph_0$ then $|X-Y|=2^{\aleph_0}$ without using axiom of choice or continuum hypothesis. Most of the proofs I'...
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### Which sets have nontrivial elementary extensions (with respect to all possible relations) of the same cardinality?

Let $\kappa$ be an infinite cardinal, and take a set $X$ of cardinality $\kappa$. Consider $X$ as a first-order structure with respect to the language that consists of all possible finitary relations ...